Average Error: 15.4 → 0.0
Time: 1.8s
Precision: 64
\[\frac{x + y}{\left(x \cdot 2\right) \cdot y}\]
↓
\[\mathsf{fma}\left(0.5, \frac{1}{y}, 0.5 \cdot \frac{1}{x}\right)\]
\frac{x + y}{\left(x \cdot 2\right) \cdot y}
↓
\mathsf{fma}\left(0.5, \frac{1}{y}, 0.5 \cdot \frac{1}{x}\right)
double f(double x, double y) {
        double r510124 = x;
        double r510125 = y;
        double r510126 = r510124 + r510125;
        double r510127 = 2.0;
        double r510128 = r510124 * r510127;
        double r510129 = r510128 * r510125;
        double r510130 = r510126 / r510129;
        return r510130;
}
↓
double f(double x, double y) {
        double r510131 = 0.5;
        double r510132 = 1.0;
        double r510133 = y;
        double r510134 = r510132 / r510133;
        double r510135 = x;
        double r510136 = r510132 / r510135;
        double r510137 = r510131 * r510136;
        double r510138 = fma(r510131, r510134, r510137);
        return r510138;
}

Error

Bits error versus x

Bits error versus y

Target

Original15.4
Target0.0
Herbie0.0
\[\frac{0.5}{x} + \frac{0.5}{y}\]

Derivation

  1. Initial program 15.4

    \[\frac{x + y}{\left(x \cdot 2\right) \cdot y}\]

  2. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{0.5 \cdot \frac{1}{y} + 0.5 \cdot \frac{1}{x}}\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{1}{y}, 0.5 \cdot \frac{1}{x}\right)}\]

  4. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(0.5, \frac{1}{y}, 0.5 \cdot \frac{1}{x}\right)\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x y)
  :name "Linear.Projection:inversePerspective from linear-1.19.1.3, C"
  :precision binary64

  :herbie-target
  (+ (/ 0.5 x) (/ 0.5 y))

  (/ (+ x y) (* (* x 2) y)))